runner on 3B, 2-1 count on batter, batter swings, drop by catcher. Batter thinks it's 3rd strike and runs to first. Catcher throws to 1B and run scores from 3rd.

We allowed run to score, brought batter back to plate with 2-2 count

How many outs?

RE: The batter swung and missed for strike 2 and then left the box to run towards first base. This is a violation of FED's batter's box rule. In the rule they list the reasons when B1 is allowed to vacate the box. Thinking it was strike 3 instead of strike 2 is NOT one of them.

We have strike 3 on the batter for leaving the box area.

Now it depends upon how many outs there are to determine if R3's run counts or not.

Less then 2 outs we have B1 out on strike 3 and R3's run counts

If there were 2 outs then we have strike 3 on the batter and no Runs score.